Related papers: Automated solving of constant-coefficients second-…
This work obtains a fixed-point equation for the solution of linear parabolic partial differential problems based on solutions to heat problems. This is a pointwise equality, so we have required non-standard techniques that involve the…
We have been working in many aspects of the problem of analyzing, understanding and solving ordinary differential equations (first and second order). As we have extensively mentioned, while working in the Darboux type methods, the most…
The inverse problem of determining the order of the fractional Riemann- Liouville derivative with respect to time in the subdi_usion equation with an arbitrary positive self-adjoint operator having a discrete spectrum is considered. Using…
Heat fluctuations over a time \tau in a non-equilibrium stationary state and in a transient state are studied for a simple system with deterministic and stochastic components: a Brownian particle dragged through a fluid by a harmonic…
To broaden the range of applicability of variable-order fractional differential models, reliable numerical approaches are needed to solve the model equation. In this paper, we develop Laguerre spectral collocation methods for solving…
In the last three decades, Fourier analysis methods have known a growing importance in the study of linear and nonlinear PDE's. In particular, techniques based on Littlewood-Paley decomposition and paradifferential calculus have proved to…
The solutions of fractional differential equations (FDEs) have a natural singularity at the initial point. The accuracy of their numerical solutions is lower than the accuracy of the numerical solutions of FDEs whose solutions are…
Pseudospectral numerical schemes for solving the Dirac equation in general static curved space are derived using a pseudodifferential representation of the Dirac equation along with a simple Fourier-basis technique. Owing to the presence of…
Recent work in deep learning focuses on solving physical systems in the Ordinary Differential Equation or Partial Differential Equation. This current work proposed a variant of Convolutional Neural Networks (CNNs) that can learn the hidden…
This paper considers the initial-boundary value problem for the heat equation with a dynamic type boundary condition. Under some regularity, consistency and orthogonality conditions, the existence, uniqueness and continuous dependence upon…
We develop a well-posedness theory for second order systems in bounded domains where boundary phenomena like glancing and surface waves play an important role. Attempts have previously been made to write a second order system consisting of…
The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases…
In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semi-linear PDEs in a Hilbert space $H^{l}\subset H^{s}(\mathbb{R}^{m})$ ($s\geq1$) via computer-assisted proofs.…
This paper introduces a fast and numerically stable algorithm for the solution of fourth-order linear boundary value problems on an interval. This type of equation arises in a variety of settings in physics and signal processing. Our method…
Neural operators have emerged as a powerful tool for solving partial differential equations in the context of scientific machine learning. Here, we implement and train a modified Fourier neural operator as a surrogate solver for…
We consider the symmetry-breaking steady state bifurcation of a spatially-uniform equilibrium solution of E(2)-equivariant PDEs. We restrict the space of solutions to those that are doubly-periodic with respect to a square or hexagonal…
This paper presents an operational framework for the computation of the discretized solutions for relativistic equations of Klein-Gordon and Dirac type. The proposed method relies on the construction of an evolution-type operador from the…
This paper explores the feasibility of quantum simulation for partial differential equations (PDEs) with physical boundary or interface conditions. Semi-discretisation of such problems does not necessarily yield Hamiltonian dynamics and…
A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of…
Recently, the numerical schemes of the Fokker-Planck equations describing anomalous diffusion with two internal states have been proposed in [Nie, Sun and Deng, arXiv: 1811.04723], which use convolution quadrature to approximate the…